A256325 a(n) = Sum_{k=0..n-1} (n-k)!*exp(-k/2)*M_{k-n,1/2}(k), where M is the Whittaker function.

0, 0, 1, 5, 24, 136, 933, 7589, 71376, 760796, 9051353, 118784325, 1703388648, 26486926720, 443732646029, 7965563713781, 152504645563072, 3101366761047860, 66753627906345057, 1515914174890163541, 36218232449903567992, 908098606824551207384, 23839591584412453131765

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

Index entries for sequences related to Laguerre polynomials

Formula

a(n) = Sum_{k=0..n-1} k*(n-k)!*hypergeom([k-n+1],[2],-k).

a(n) = Sum_{k=0..n-1}(Sum_{j=0.. n-k}((n-k-j)!*C(n-k,j)*C(n-k-1,j-1)*k^j)).

a(n) = Sum_{k=0..n-1} (n-k-1)* k! * LaguerreL(k, 1, k-n+1). - G. C. Greubel, Feb 23 2021

Maple

a := n -> add(exp(-k/2)*WhittakerM(-(n-k), 1/2, k)*(n-k)!, k=0..n-1):
seq(round(evalf(a(n), 64)), n=0..22);
# Alternatively:
a := n -> add(k*(n-k)!*hypergeom([k-n+1], [2], -k), k=0..n-1):
seq(simplify(a(n)), n=0..22);

Mathematica

Table[Sum[(n-k-1)*k!*LaguerreL[k, 1, k-n+1], {k, 0, n-1}], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)

Prog

(Sage) [sum( (n-k-1)*factorial(k)*gen_laguerre(k, 1, k-n+1) for k in (0..n-1) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021
(Magma) [n eq 0 select 0 else (&+[(n-k-1)*Factorial(k)*Evaluate( LaguerrePolynomial(k, 1), k-n+1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

Crossrefs

Cf. A253286.

Sequence in context: A009411 A080996 A020055 * A286743 A232318 A201952

Adjacent sequences: A256322 A256323 A256324 * A256326 A256327 A256328

Keyword

nonn,easy

Author

Peter Luschny, Mar 24 2015

Status

approved